When duplicate groups the copy is in the same group
The reason that this is happening is that when you duplicate an object all of its properties are duplicated as well, not just the geometry. Group membership is one of those just like materials and modifiers. There isn't a way to copy only a limited portion of an objects properties so it's all or nothing. The shortcut provided by Dan will work to remove the object from the group. Remember that you can add and remove multiple object from groups with Alt+G to add to a group and Ctrl+Alt+G to remove from group. The name of the group can be changed in the outliner.

Update: In response to Kim Holder's statement about 'Alt+G'. That is correct and what I should have put was 'Ctrl+G' to add to a group. When you do that the Operator window in the Tools panel will give you the option of naming the group or you can change the name in the outliner.